Mathematical Methods for Physicists: A concise introduction - Site Map

COMPLEX INTEGRATION
Solution: For convenience, let z z 0 ˆ re i , where ranges from 0 to 2 as z
ranges around the circle (Fig. 6.7). Then dz ˆ rie i d, and the integral becomes
If n ˆ 0, this reduces to
and if n 6ˆ 0, we have
Z 2
0
rie i d
r n‡1 e ˆ i Z 2
i…n‡1† r n
i
Z 2
0
0
d ˆ 2i
e in d:
Z
i 2
r n …cos n i sin n†d ˆ 0:
0
This is an important and useful result to which we will refer later.
Cauchy's integral theorem
Cauchy's integral theorem has various theoretical and practical consequences. It
states that if f …z† is analytic in a simply-connected region (domain) and on its
boundary C, then
I
f …z†dz ˆ 0:
…6:25†
C
What do we mean by a simply-connected region? A region R (mathematicians
prefer the term `domain') is called simply-connected if any simple closed curve
which lies in R can be shrunk to a point without leaving R. That is, a simplyconnected
region has no hole in it (Fig. 6.7(a)); this is not true for a multiplyconnected
region. The multiply-connected regions of Fig. 6.7(b) and (c) have
respectively one and three holes in them.
Figure 6.7.
Simply-connected and doubly-connected regions.
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